Variational bounds on some bulk properties of a two-phase composite material

A variational principle due to Hashin and Shtrikman is used to obtain theoretical upper and lower bounds on the effective bulk dielectric constant εe (or an analogous property such as magnetic permeability, electrical or thermal conductivity, or a diffusivity) of a two-phase macroscopically homogeneous composite material from information about another similar effective bulk property. For the case of a composite whose macroscopic symmetry under rotations is either isotropic or cubic, we obtain a new and rather simple pair of bounds that are usually considerably better than any of those that are presently obtainable under these conditions.